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 Incoming links: Exponential transform of a standard Gumbel variable

Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 9 / a / iv / Solution 2026-10-25

Because

Z = e^{- Y} > 0

, its cumulative distribution function is zero for

z \leq 0

. For

z > 0

, the transformation is decreasing, and

Y

has a continuous cumulative distribution function, so

P (Z \leq z) = P (Y \geq - ln z) = 1 - F (- ln z) = 1 - e^{- z} .

Thus the exponential transform of a standard Gumbel variable gives

F_{Z} (z) = {0, 1 - e^{- z}, z \leq 0, z > 0, Z \sim Exp (1) .

This is the exponential distribution with rate

1

.

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